Here we briefly recap the open-boundary winding numbers (OBWNs) introduced in Ref. [1]}.
They will be used to establish the bulk-edge correspondence for the first parent model in Sec. 
and its \(q\) th-root descendants. We first consider the dynamics of the model in two symmetric time frames,
where the Floquet operator \(U=e^{-iH_1/2}e^{-iH_2/2}\)  is transformed to \(U_a=e^{-iH_2/4}e^{-iH_1/2}e^{-iH_2/4}\)  and
\(U_b=e^{-iH_1/4}e^{-iH_2/2}e^{-iH_1/4}\) . Next we define the \(Q\) -matrix in the time frame \(\alpha \)  (\(=a,b\) ) as
\(Q_\alpha =\sum _j(|\psi ^+_{\alpha j}\rangle \langle {\tilde{\psi }}^+_{\alpha j}|-|\psi ^-_{\alpha j}\rangle \langle {\tilde{\psi }}^-_{\alpha j}|)\) .
The right and left Floquet eigenvectors \(|\psi ^\pm _{\alpha j}\rangle \)  and \(\langle {\tilde{\psi }}^\pm _{\alpha j}|\)  satisfy the eigenvalue equations
\(U_\alpha |\psi ^\pm _{\alpha j}\rangle =e^{-i(\pm E_j)}|\psi ^\pm _{\alpha j}\rangle \) 
and \(\langle {\tilde{\psi }}^\pm _{\alpha j}|U_\alpha =\langle {\tilde{\psi }}^\pm _{\alpha j}|e^{-i(\pm E_j)}\) .
An OBWN for \(U_\alpha \)  is then defined as \(\nu _\alpha ={\rm Tr_B}(\Gamma Q_\alpha [Q_\alpha ,X])/L_{\rm B}\) .
Here \(\Gamma \)  is the CS operator. \(X\)  is the unit cell position operator.
For a system with \(L\)  lattice sites, we decompose it into a bulk region and two edge regions
at the left and right. The trace \({\rm Tr_B}\)  is taken over the bulk region, which contains \(L_{\rm B}\)  lattice sites.
The length of each edge region is \(L_{E}=(L-L_{\rm B})/2\) , which should be chosen properly in order to avoid the
obstruction of non-Hermitian skin effect.
Finally, we define two OBWNs for a 1D non-Hermitian FTI with CS as \(\nu _0=(\nu _a+\nu _b)/2\)  and \(\nu _0=(\nu _a-\nu _b)/2\) .
These winding numbers can only take integer values. They are further related to the numbers of Floquet edge modes at zero and \(\pi \) 
quasienergies \(n_0\)  and \(n_\pi \)  through the relations \((n_0,n_\pi )=2(|\nu _0|,|\nu _\pi |)\) . Following our analysis in the Sec. 
of the main text, \((\nu _0,\nu _\pi )\)  could thus also count the numbers of fractional-quasienergy edge modes in the \(q\) th-root descendants of
the parent model \(U\) .
